Determining Factorial Speed Fast

TL;DR

This paper establishes that graph classes representable by finite binary languages have at most factorial speed, classifies many previously unknown classes as factorial, and shows that k-letter graphs have exponential speed.

Contribution

It introduces a criterion linking finite binary language representation to factorial speed and classifies numerous graph classes accordingly, clarifying inclusion relations.

Findings

Graph classes with finite binary language representation have at most factorial speed.

Many previously unknown graph classes are now classified as factorial.

k-letter graphs have exponential speed.

Abstract

The speed of a graph class $\cal G$ measures how many labeled graphs on $n$ vertices one can find in $\cal G$. This graph class complexity function is explicitly provided on graphclasses.org. However, for many graph classes, their speed status is classified as \emph{unknown}. In this paper, w}\shortversion{W}e show that any graph class representable by a finite binary language has at most factorial speed, meaning that its speed function behaves like $2^{\Theta(n\log n)}$, and we use this criterion to classify many graph classes whose speed was previously unknown as factorial. As a consequence, inclusions between several graph classes can now be seen to be proper. We also prove that $k$-letter graphs have exponential speed, i.e., the speed function lies in $2^{\Theta(n)}$.